TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/source_sewing_20261029/PROOF.txt Original file SHA-256: 9b6ed0c36862913dac9762c0d1a6cd1e9f9c9a5f6b4b905ac8de5dccc25424b1 Inclusive original lines: 1-184 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | FIRST SOURCE SEWING TEST: A SPECTRAL WEIGHT IS NOT AN OS POSITIVITY PROOF 000002 | 29 October 2026. Continuation of the existing original-source investigation. 000003 | 000004 | Requested result 000005 | 000006 | Construct actual supported P1 boundary insertions and derive their joint 000007 | functional, reflection and time, reusing the existing microscopic/local source 000008 | maps. The physical source map has NOT been constructed in this note. The first 000009 | reviewed shortcut through an old measure theorem fails at a precise inference. 000010 | This exclusion is a contribution to that provenance question, not its closure. 000011 | 000012 | 1. Actual source and old results retained 000013 | 000014 | The original primitive action is a relative spectral trace with a declared 000015 | profile, reference and eta data. Original archive45 boundary source001:302-316 000016 | does not define its fermionic source coupling. Archive45 QFT source002:256-310 000017 | defines Z_rel[J,eta,bareta]=Z_adm[J,eta,bareta]/Z_ref[0] and explicitly assumes 000018 | that the admissible fermionic CAR kernel is reflection positive. 000019 | 000020 | The existing free Schur integration is already solved, see Atlas finding 000021 | joint-fermionic-source-schur-functional and seam-collar-multifield-20261016/ 000022 | PROOF.txt:271-326. It assumes K, the fermion measure and its source coupling. 000023 | It is reused, not a new derivation of these inputs from a spectral trace. 000024 | The existing two_face_source_20261029/PROOF.txt already has X, all four graph 000025 | blocks, the graded HS partner, bounded neutral responses and heat-product OS 000026 | forms on the given source. None is reselected or invalidated by this note. 000027 | 000028 | The old archive45 metrology source005:136-179 claims a constructive geometric 000029 | measure with density exp[-Tr f_star(D_V^2/Lambda^2)]. Its proof:170-173 says 000030 | that the preceding OS positivity is preserved by the positive spectral weight. 000031 | The matched archive42 source016:108-151 has the same step. Both have additional 000032 | geometric/BRST/exhaustion assumptions which the following finite example does 000033 | NOT instantiate. We test only the purported general preservation principle. 000034 | The earlier failure of the UV fixed-point map is already documented and is 000035 | not counted as a new result here. 000036 | 000037 | 2. Exact finite reflected-measure test 000038 | 000039 | There are two halves, each carrying a sign s,t in {-1,+1}. Reflection exchanges 000040 | them and conjugates scalars. The positive-half algebra is all functions F(t). 000041 | An unnormalized symmetric joint measure is the matrix K_st. Its reflected form 000042 | is sum_st conjugate(F(s)) K_st F(t), divided by sum_st K_st for normalization. 000043 | Thus reflection positivity is exactly K positive semidefinite. 000044 | 000045 | Use the base kernel K0=[[10,9],[9,10]]. All entries are positive, both eigenvalues 000046 | are positive (19,1), and the associated normalized transfer is primitive. For 000047 | F(t)=t, the original normalized reflected form is 1/19>0. This is a strictly RP 000048 | base; the example does not exploit a null vector of a merely semidefinite base. 000049 | 000050 | For a>g>0 define the self-adjoint matrix 000051 | D(s,t)=[[a s,g],[g,a t]]. 000052 | With R=[[0,1],[1,0]] it obeys R D(s,t) R=D(t,s). Its squared spectra are 000053 | s=t: (a-g)^2,(a+g)^2; 000054 | s=-t: a^2+g^2,a^2+g^2. 000055 | At a=2,g=1 every matrix is invertible with absolute spectral gap at least1. 000056 | This is a finite spectral test, not an elliptic geometric/BRST realization. 000057 | 000058 | Take f(u)=exp(-u), a positive completely monotone heat profile with positive 000059 | Laplace measure delta_1, in the class called the Bernstein cone by the source. 000060 | It is not a Bernstein function in the separate conventional increasing sense. 000061 | For S(s,t)=Tr f(D(s,t)^2), q=exp(-1), one gets 000062 | S_same=q+q^9, 000063 | S_opposite=2q^5, 000064 | Delta=S_same-S_opposite=q(1-q^4)^2>0. 000065 | The Boltzmann weights w(s,t)=exp[-S(s,t)] are strictly positive and invariant 000066 | under the reflection. The weighted reflected kernel is nonetheless 000067 | K=[[10 exp(-S_same),9 exp(-S_opposite)], 000068 | [9 exp(-S_opposite),10 exp(-S_same)]]. 000069 | Its eigenvalue in the F(t)=t direction is 000070 | lambda_minus=exp(-S_opposite)[10 exp(-Delta)-9]. 000071 | 000072 | This sign is rigorously negative. The elementary series for e gives 2(1/3)(15/16)^2=75/256>1/9>log(10/9). 000075 | Consequently 10 exp(-Delta)<9. Normalization by a positive partition function 000076 | cannot change that sign. Explicitly, 000077 | E_w[(Theta F)F] 000078 | =[10 exp(-S_same)-9 exp(-S_opposite)] 000079 | /[10 exp(-S_same)+9 exp(-S_opposite)] 000080 | =-0.1239426461250916662535013947009952773381... 000081 | The number is a value of THIS counterexample, not a computed TFPT response. 000082 | Every configuration has positive ordinary probability, but the reflected form 000083 | is negative. Pointwise positivity, reflection symmetry and a spectral gap do 000084 | not suffice for preservation of OS positivity under this reweighting. 000085 | 000086 | 3. What generalizes, and what does not 000087 | 000088 | For any finite positive Laplace measure nu, 000089 | f(u)=integral exp(-r u) dnu(r), 000090 | the same squared spectra give 000091 | Delta=2 integral exp[-r(a^2+g^2)] 000092 | [cosh(2rag)-1] dnu(r). 000093 | This is strictly positive when nu((0,infinity))>0. In particular the issue is 000094 | not caused by negative heat weights. For the fixed base9/10 one still needs 000095 | Delta>log(10/9); arbitrarily small profiles need not violate its RP condition. 000096 | 000097 | For every such nonconstant profile, choosing a strictly RP base 000098 | K0_rho=[[1,rho],[rho,1]], exp(-Delta)=0 for every 000108 | K0>=0. Conversely, a multiplier preserving every such K0 must have W>=0 by 000109 | testing K0=all-ones; strict positive bases follow by small diagonal perturbation. 000110 | Thus pointwise positive entries of W must not be substituted for W>=0. 000111 | For a particular fixed base, K0 elementwise-times W>=0 is the exact criterion; 000112 | W>=0 alone is sufficient, not necessary for that particular base. 000113 | 000114 | For bounded functions on a classical reflected configuration space, the 000115 | factorization 000116 | w=sum_j c_j Theta(B_j) B_j, c_j>=0, B_j on the positive half, 000117 | gives q_w(F)=sum_j c_j q_0(B_j F)>=0. Controlled integrable limits preserve the 000118 | same conclusion. A factorization exp[-S_plus-Theta(S_plus)] is a special case. 000119 | Odd fermionic products require their actual graded/twisted conventions; the 000120 | commutative proof is not silently extended to arbitrary Grassmann densities. 000121 | 000122 | For the actual P1 claim, one must derive such a suitable sewing structure from 000123 | the original full admissible action/measure, or directly prove positivity of 000124 | its weighted reflected kernel. A positive spectral trace alone supplies none. 000125 | No compensating coupling, new profile or preferred state is chosen here. 000126 | 000127 | 5. Relative normalization does not determine the missing source response 000128 | 000129 | When Z_ref[0] is a fixed positive source-independent denominator, 000130 | Z_rel[J]/Z_rel[0]=Z_adm[J]/Z_adm[0], 000131 | delta_L delta_R log Z_rel=delta_L delta_R log Z_adm. 000132 | The statement applies with the same fixed order of Grassmann derivatives. 000133 | The original denominator preserves an ALREADY positive admissible reflected 000134 | form; it cannot make a negative form positive or supply a missing cross term. 000135 | This does not remove reference-dependent terms from the numerator action, 000136 | nor authorize changing the fixed determinant family or counterterms. 000137 | 000138 | 6. Relation to existing verification and the actual local target 000139 | 000140 | v330_qgamb_admissible_measure.py:42-50 constructs a specified PSD transfer from 000141 | three chosen eigenvectors/eigenvalues. Its finite transfer positivity is a 000142 | valid given-source fact; run:60-132 explicitly leaves the metric/coupling part 000143 | open. That construction does not certify an arbitrary positive spectral 000144 | reweighting. It is retained rather than replaced by the finite counterexample. 000145 | 000146 | The old measure theorem therefore cannot close the actual P1 source bridge 000147 | using the tested preservation inference alone. A direct local neutral source 000148 | embedding may still avoid constructing the whole ambient geometric measure. 000149 | Full ambient measure construction is NOT introduced as a new prerequisite for 000150 | every neutral response. The target remains the actual supported local map and 000151 | its joint functional with the correct products, phases, reflection and time. 000152 | The old KMS/HS reference response remains the explicit comparison functional. 000153 | 000154 | Next decisive physical obligation: identify the actual admissible insertion 000155 | map and show that its normalized joint generating coefficients agree on the 000156 | needed local neutral histories, using an original-source sewing law or a 000157 | direct local OS proof. Neither a formal name Z_adm nor the pointwise spectral 000158 | weight proves that equality. If the original interacting source differs from 000159 | the free projector, its actual higher joint words must be retained. 000160 | 000161 | 7. Verification and primary literature 000162 | 000163 | check_spectral_weight_rp.py checks exact spectra, gaps, reflection covariance, 000164 | the polynomial Delta identity and rational sign bound. It independently 000165 | computes all four matrix heat traces and the normalized observable form, and 000166 | checks cancellation of a source-independent reference. Numerical evaluation 000167 | only illustrates the exact analytic sign proof. INDEPENDENT-REVIEW.txt reviews 000168 | the mathematics and its limited application to the original inference. 000169 | 000170 | Ziebell, theta-splitting densities and reflection positivity (2024), Definition 000171 | 3.4 and Theorem3.5, requires theta-splitting/integrability beyond positivity of 000172 | a density, in its stated Gaussian distribution-space setting: 000173 | https://doi.org/10.1007/s11005-024-01799-8 000174 | Jaffe and Janssens, Reflection Positive Doubles (2017), gives reflected-cone 000175 | and coupling criteria, not an unrestricted positive-density preservation rule: 000176 | https://arxiv.org/abs/1607.07126 000177 | Our finite proof above is elementary and does not require extending either 000178 | paper beyond its hypotheses. No claim of a new general RP theorem is made. 000179 | 000180 | Verdict: the generic positive-spectral-weight preservation inference is 000181 | EXCLUDED_SCOPED. The actual physical P1 field/sewing map remains unproved. 000182 | No full P1/P2 countermodel, no TFPT physical prediction, no new primitive 000183 | model, no full continuum measure or T1-T8 closure is asserted. 000184 | complete_TFPT_solution=false; ledger_changed=false.